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My teacher gave us 50 possible test questions of which he'll randomly select 25 to test us on. From the 25, we can then select 10 to do of which he'll grade the 8 best.

The question is how many problems can I NOT study (of the 50) and still be guaranteed a 100 on the test? (assuming the ones I study I do so perfectly)

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By using the word 'guarantee' this ceases to be a probability problem; it's simply a matter of finding the worst-case.

Consider if you only studied 25 of them. He could - with low probability - randomly select the 25 you didn't study. You might get zero!

If you studied 25+8, those extra 8 must appear on the test and you'll get 100% on those. As long as those 8 are in your 10 you get 100%.

So 33 is sufficient.

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You should study because you want to learn, not because you want to pass a test! And get off my lawn.

That aside, if you want to a guarantee, then this isn't about probability, since you need 100% probability to guarantee something. In your case, the worst possible situation is that 25 of the questions you've studied are in the 'not selected' pile. For the remaining 25, you can select the ones you want and you need to be able to answer 8 of them. That makes the total number you need to study 25 + 8 = 33.

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